Rate of Convergence for a Nonlocal-to-local Limit in One Dimension
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916732025700352 |
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| author | Carrillo, José A. Elbar, Charles Fronzoni, Stefano Skrzeczkowski, Jakub |
| author_facet | Carrillo, José A. Elbar, Charles Fronzoni, Stefano Skrzeczkowski, Jakub |
| contents | We consider a nonlocal approximation of the quadratic porous medium equation where the pressure is given by a convolution with a mollification kernel. It is known that when the kernel concentrates around the origin, the nonlocal equation converges to the local one. In one spatial dimension, for a particular choice of the kernel, and under mere assumptions on the initial condition, we quantify the rate of convergence in the 2-Wasserstein distance. Our proof is very simple, exploiting the so-called Evolutionary Variational Inequality for both the nonlocal and local equations as well as a priori estimates. We also present numerical simulations using the finite volume method, which suggests that the obtained rate can be improved - this will be addressed in a forthcoming work. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07015 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rate of Convergence for a Nonlocal-to-local Limit in One Dimension Carrillo, José A. Elbar, Charles Fronzoni, Stefano Skrzeczkowski, Jakub Analysis of PDEs Numerical Analysis 35A15, 35Q70, 35B40, 65M08 We consider a nonlocal approximation of the quadratic porous medium equation where the pressure is given by a convolution with a mollification kernel. It is known that when the kernel concentrates around the origin, the nonlocal equation converges to the local one. In one spatial dimension, for a particular choice of the kernel, and under mere assumptions on the initial condition, we quantify the rate of convergence in the 2-Wasserstein distance. Our proof is very simple, exploiting the so-called Evolutionary Variational Inequality for both the nonlocal and local equations as well as a priori estimates. We also present numerical simulations using the finite volume method, which suggests that the obtained rate can be improved - this will be addressed in a forthcoming work. |
| title | Rate of Convergence for a Nonlocal-to-local Limit in One Dimension |
| topic | Analysis of PDEs Numerical Analysis 35A15, 35Q70, 35B40, 65M08 |
| url | https://arxiv.org/abs/2505.07015 |